Is it possible to find the complex roots of $x^3 + 2 x^2 - 3 = 0$

Question

Sorry to ask so many roots questions in such short bursts, but I want to know if it is humanly possible to compute the complex root of

$$x^3 + 2 x^2 - 3 = 0$$

through algebraic manipulation?

Note that it also has a real root if you factor it a little bit

If not possible, please tell me how you can arrive at that conclusion

Thank you!

Answer 1

Note by the Rational Root Theorem that $1$ is a root of the cubic. Proceeding by polynomial long division gives the factorization $$(x-1)(x^2+3x+3)$$ You can then use the quadratic formula and find the roots of the second factor.

Answer 2

$x^3+2x^2-3=x^2(x-1)+3(x-1)(x+1)=(x-1)(x^2+3x+3)$.

So, the roots of the original equation are $x=1$ and the two roots of $x^2+3x+3$.

You can find the roots of $x^2+3x+3$ by using the quadratic formula.

Answer 3

As we see that $x=1$ is a root of the equation, we therefore factor it into $(x-1)(x^2+3x+3)$. Now, with the quadratic formula, the roots are: $$\begin{align} x=&\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =&\frac{-3\pm \sqrt{9-12}}{2}\\ =&\frac{-3\pm i\sqrt 3}{2} \end{align}$$